Cancellation of projective modules over non-Noetherian rings

TL;DR

This paper investigates conditions under which projective modules over certain polynomial and Laurent polynomial rings over rings of dimension d are cancellative, extending known results to non-Noetherian contexts.

Contribution

It generalizes the cancellation property of projective modules to non-Noetherian rings for polynomial and Laurent polynomial extensions, under specific rank and condition constraints.

Findings

E(A⊕P) acts transitively on Um(A⊕P) for modules P of rank ≥ d+1

Extends known results to non-Noetherian rings

Provides conditions for cancellation of projective modules over polynomial rings

Abstract

Let R be a ring of dimension d and A be one of R[Y] or R[Y,Y^{-1}]. If P is a projective A-module of rank \geq d+1 satisfying some condition, then we show that E(A\oplus P) acts transitively on Um(A\oplus P). When P is free, this result is due to Yengui (when A=R[Y]) and Abedelfatah (when A=R[Y,Y^{-1}]).